Simulation and the monte carlo method

Computational methods in optimization considering uncertainties – An overview

Stochastic Optimal Control

Reliability-based Optimization is a most appropriate and advantageous methodology for structural design. Its main feature is that it allows determining the best design solution (with respect to prescribed criteria) while explicitly considering the unavoidable effects of uncertainty. In general, the application of this methodology is numerically involved, as it implies the simultaneous solution of an optimization problem and also the use of specialized algorithms for quantifying the effects of uncertainties. In view of this fact, several approaches have been developed in the literature for applying this methodology in problems of practical interest. This contribution provides a survey on approaches for performing Reliability-based Optimization, with emphasis on the theoretical foundations and the main assumptions involved. Early approaches as well as the most recently developed methods are covered. In addition, a qualitative comparison is performed in order to provide some general guidelines on the range of applicability on the different approaches discussed in this contribution.

A survey on approaches for reliability-based optimization

The aerodynamic characteristics of train/vehicles under cross winds depend on not only the shapes of the vehicles but also those of infrastructures. Accordingly three kinds of wind tunnel tests were made to evaluate the aerodynamic characteristics of typical configurations of the vehicles on typical configurations of infrastructures such as bridges and embankments. The main results obtained from the wind tunnel tests are summarized below. 1. The aerodynamic side force coefficient of the vehicle increases more as the thickness of the bridge girder becomes larger. It also increases more as the roof of the vehicle becomes edgier. 2. The aerodynamic characteristics of the vehicle on the embankment depend on the distribution of the boundary layer on the ground. The aerodynamic side force coefficient of the vehicle on a high embankment is larger than that on a low embankment.

Aerodynamic Characteristics of Train/Vehicles under Cross Winds

This paper presents a review of recent research that has been carried out on the cross-wind effects on road and rail vehicles. After a brief introduction to the issues involved, the risk analysis framework is set out. All risk analysis methods require some knowledge of cross-wind aerodynamic force and moment coefficients, and methods of obtaining these through full scale and wind tunnel testing and through Computational Fluid Dynamics methods are then described. The picture of the flow fields around vehicles that is suggested by these measurements and calculations is then presented, and the steady and the unsteady aerodynamic force characteristics described. The detailed methodology for using this information to predict accident risk is then set out, including details of the vehicle dynamics system models that can be used. Finally potential alleviation methods are described and suggestions made for further works.

Cross-wind effects on road and rail vehicles

Equivalent linearization and Monte Carlo simulation in stochastic dynamics

Like with other types of fluid bearings, rotors supported by floating ring bearings may become unstable with increasing speed of rotation due to self-excited vibrations. In order to study the effects of the nonlinear bearing forces, within this contribution a perfectly balanced symmetric rotor is considered which is supported by two identical floating ring bearings. Here, the bearing forces are modeled by applying the short bearing theory for both fluid films. A linear stability analysis about the static equilibrium position of the rotor shows that for a critical revolution speed the real part of an eigenvalue pair changes its sign. By means of a center manifold reduction it is shown that this destabilization of the steady state is due to a Hopf-bifurcation. Furthermore, the type of this bifurcation is determined as well as the existence and stability of limit-cycles. Notably it is found that depending on the parameters of the floating ring bearing subcritical as well as supercritical bifurcations may occur. Additionally, the analytical results obtained from the center manifold reduction are compared to numerical results by a continuation method. In conclusion, the influences of bearing design parameters on the stability and on the limit-cycles are discussed.

Analytical bifurcation analysis of a rotor supported by floating ring bearings

The paper presents exact stationary probability density functions for systems under Poisson white noise excitation Two different solution methods are outlined In the first one, a class of non-linear systems is determined whose state vector is a memoryless transformation of the state vector of a linear system The second method considers the generalized Fokker–Planck (Kolmogorov-forward) equation Non-linear system functions are identified such that the stationary solution of the system admits a prescribed stationary probability density function Both methods make use of the stochastic integro-differential equations approach This approach seems to have some computational advantages for the determination of exact stationary probability density functions when compared to the stochastic differential equations approach

Exact stationary probability density functions for non-linear systems under Poisson white noise excitation

Several stochastic linearization techniques are derived for nonlinear systems under parametrical Poisson white noise excitation. The differential equations for the first and second order moments of the linearized systems are obtained and differences to the corresponding moment equations of the nonlinear system are discussed. It is shown that different linear models may lead to ‘true’ linearization coefficients in the sense of Kozin in: F. Ziegler, G.I. Schueller (Eds.), Proceedings of the IUTAM Symposium on Nonlinear Stochastic Dynamic Engineering Systems, Springer, Berlin, Heidelberg, New York, 1988, pp. 45–56. For the Duffing oscillator and the van der Pol oscillator, the results are compared with Monte Carlo simulation.

Carsten Proppe

Papers

Cross-wind effects on road and rail vehicles

Equivalent linearization and Monte Carlo simulation in stochastic dynamics

Analytical bifurcation analysis of a rotor supported by floating ring bearings

Exact stationary probability density functions for non-linear systems under Poisson white noise excitation

Stochastic linearization of dynamical systems under parametric Poisson white noise excitation